Problem: Solve for $x$, ignoring any extraneous solutions: $\dfrac{x^2 + 11x}{x - 9} = \dfrac{23x - 27}{x - 9}$
Answer: Multiply both sides by $x - 9$ $ \dfrac{x^2 + 11x}{x - 9} (x - 9) = \dfrac{23x - 27}{x - 9} (x - 9)$ $ x^2 + 11x = 23x - 27$ Subtract $23x - 27$ from both sides: $ x^2 + 11x - (23x - 27) = 23x - 27 - (23x - 27)$ $ x^2 + 11x - 23x + 27 = 0$ $ x^2 - 12x + 27 = 0$ Factor the expression: $ (x - 3)(x - 9) = 0$ Therefore $x = 3$ or $x = 9$ However, the original expression is undefined when $x = 9$. Therefore, the only solution is $x = 3$.